I believe that you and I are in agreement, although your use of the
words "cut it 1' all the way" could be open to interpretation. If you
mean "cut it *so that it ends up* 1' all the way" then we are on the
same page.

Assuming your pavers run parallel to the asphalt area, I would cut the
asphalt back whatever distance would allow full rows of pavers. That way
you won't have a bunch of cut pavers along the edge.

In other words, if your pavers are 7" wide, cut back 14" so you can fit two
full rows of pavers.

Well, therein lies the problem: The backfill. What would I use to
backfill along the edge?

If you prepare it properly for pavers it shouldn't be an issue. Dig down
about 8", install about 5-6" of 5/8" minus gravel (tamping well after every
2" addition), add your setting sand, screed, install pavers. Sweep in
polymeric sand to lock the pavers together and water it in.

Ideally, I would bring the pavers tight up against the cut
edge and they would provide the lateral support required to keep the
edge from cracking. The problem is, if they don't provide enough
support when the vehicles are parked there, I'm screwed.

Is the existing edge cracking? If not, why would the new edge crack?

If you're really worried about it, remove the pavers and pour a concrete
slab to fill the new space. I would still cut the asphalt back for a
straight edge, but you could leave the current angle if you're not wanting
to mess with the asphalt at all.

Take care,

Anthony Watson
www.mountainsoftware.com
www.watsondiy.com